STATISTICS: MODULE Chapter 3 - Bivariate or joint probability distributions

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "STATISTICS: MODULE 12122. Chapter 3 - Bivariate or joint probability distributions"

Transcription

1 STATISTICS: MODULE Chapte - Bivaiate o joit pobabilit distibutios I this chapte we coside the distibutio of two adom vaiables whee both adom vaiables ae discete (cosideed fist) ad pobabl moe impotatl whee both adom vaiables ae cotiuous. Bivaiate o joit distibutios model the wa two adom vaiables va togethe. A. DISCRETE VARIABLES Eample. Hee we have a pobabilit model of the demad ad suppl of a peishable commodit. The pobabilit model/distibutio is defied as follows: Suppl of commodit (SP) Demad fo commodit (D) This is kow as a discete bivaiate o joit pobabilit distibutio sice thee ae two adom vaiables which ae "demad fo commodit (D)" ad "suppl of commodit (SP)". The sample space S cosists of 5 outcomes (d, s) whee d ad s ae the values of D ad SP. The pobabilities i the table ae joit pobabilities, amel P( D d ad SP s) o P( D d SP s) usig set otatio. Eamples Note: The sum of the 5 pobabilities is.. Joit pobabilit fuctio Suppose the adom vaiables ae ad, the the joit pobabilit fuctio is deoted b p ad is defied as follows: p P( ad ) o P( )

2 ,. Also p( ). Magial pobabilit distibutios The magial distibutios ae the distibutios of ad cosideed sepaatel ad model how ad va sepaatel fom each othe. Suppose the pobabilit fuctios of ad ae p ( ) ad p ( ) espectivel so that p ( ) P( ) ad p ( ) Also p ( ) ad p ( ). P( ) It is quite staightfowad to obtai these these fom the joit pobabilit distibutio p p, p p, sice ( ) ( ) ad ( ) ( ) I egessio poblems we ae ve iteested i coditioal pobabilit distibutios such as the coditioal distibutio of give ad the coditioal distibutio of give.4 Coditioal pobabilit distibutios The coditioal pobabilit fuctio of give is deoted b p( ) is defied as p( ) P( ) ( ) P( ) P ad ( ) p wheeas the coditioal pobabilit fuctio of give is deoted b p( ) ad defied as p( ) P( ) ( ) P( ) P ad p ( ) p p.5 Joit pobabilit distibutio fuctio The joit (cumulative) pobabilit distibutio fuctio (c.d.f.) is deoted b F(, ) ad is defied as F(, ) P( ad ) ad 0 F(, ) The magial c.d.f s ae deoted b F ( ) ad F ( ) F ( ) P( ) ad F ( ) (see Chapte, sectio. ). P( ) ad ae defied as follows

3 .6 Ae ad idepedet? If eithe (a) F(, ) F ( ). F ( ) o (b)p(, ) p ( ). p ( ) the ad ae idepedet adom vaiables. Eample. The joit distibutio of ad is (a) Fid the magial distibutios of ad. (b) Fid the coditioal distibutio of give 0. (c) Ae ad idepedet? B. CONTINUOUS VARIABLES.7 Joit pobabilit desit fuctio The joit p.d.f. is deoted b f (, ) (whee f (, ) 0 all ad ) ad defies a pobabilit suface i dimesios. Pobabilit is a volume ude this suface ad the total volume ude the p.d.f. suface is as the total pobabilit is i.e. f d d d b ad P( a b ad c d ) c a f d d As befoe with discete vaiables, the magial distibutios ae the distibutios of ad cosideed sepaatel ad model how ad va sepaatel fom each othe. Wheeas with discete adom vaiables we speak of magial pobabilit fuctios, with cotiuous adom vaiables we speak of magial pobabilit desit fuctios. Eample. A electoics sstem has oe of each of two diffeet tpes of compoets i joit opeatio. Let ad deote the adom legths of life of the compoets of tpe ad, espectivel. Thei joit desit fuctio is give b ( ) / f (, ) e ; + > 0 > othewise

4 4 Eample.4 The adom vaiables ad have a bivaiate omal distibutio if ( ) f, ae b whee a π σ σ ρ ad b ( ρ ) µ µ µ µ ρ + σ σ σ σ whee < <, < <, < µ <, < µ <, < ρ <, σ > 0, σ > 0. The p.d.f. suface is show below ad as ou ca see is bell-shaped..8 Magial pobabilit desit fuctio The magial p.d.f of is defied as f ( ) ad is the equatio of a cuve called the p.d.f. cuve of. P( a b) is a aea ude the p.d.f. cuve ad so P( a b) f ( ) b d (as i Chapte, sectio.8). a It ca be obtaied fom the joit p.d.f. b a sigle itegatio, as follows: f ( ) f d. The magial p.d.f of is defied as f ( ) ad is the equatio of a cuve called the p.d.f. cuve of. P( c d) is a aea ude the p.d.f. cuve ad so d. P( c d) f ( ) d c It ca be obtaied fom the joit p.d.f. b a sigle itegatio, as follows: ad f ( ) f d

5 .9 Coditioal pobabilit desit fuctios The coditioal p.d.f of give is deoted b f ( ) ad defied as f ( ) f ( ) 5 ( ) f wheeas the coditioal p.d.f of give is deoted b f ( ) ad defied as f ( ) f ( ) f ( ) f f.0 Joit pobabilit distibutio fuctio As i.5 the joit (cumulative) pobabilit distibutio fuctio (c.d.f.) is deoted b F(, ) ad is defied as F(, ) P( ad ) but F(, ) i the cotiuous case is the volume ude the p.d.f. suface fom to ad fom to, so that F( ) v u, v u f u v du dv The magial c.d.f. s ae defied as i.5 ad ca be obtaied fom the joit distibutio fuctio F(, ) as follows: F F ( ) F( MA ) ( ) F( ), whee MA is the lagest value of ad MA, whee MA is the lagest value of.. Impotat coectios betwee the p.d.f s ad the joit c.d.f. s. (i) The joif p.d.f. f (, ) F (ii) The magial p.d.f s ca be obtaied fom the magial c.d.f. s as follows: the magial p.d.f. of f ( ) the magial p.d.f. of f ( ) df d df d ( ) ( ) o F ( ), o F ( ). Ae ad idepedet? ad ae idepedet adom vaiables if eithe (a) F(, ) F ( ) F ( ) ; o

6 (b) f(, ) f ( ) f ( ) ; o 6 (c) f ( ) fuctio of ol o equivaletl f ( ) fuctio of ol Eample.5 The joit distibutio fuctio of ad is give b F + 0, 0 othewise (i) Fid the magial distibutio ad desit fuctios. (ii) Fid the joit desit fuctio. (iii) Ae ad idepedet adom vaiables? Eample.6 ad have the joit pobabilit desit fuctio 8 f, 7 (a) Deive the magial distibutio fuctio of. (b) Deive the coditioal desit fuctio of give (c) Ae ad idepedet? Give: Give: Joit desit f. f (,) Joit distibutio f. F(, ) Itegate w..t Diffeetiate (patiall) ad w..t. ad Joit distibutio f. F(, ) Joit desit f. f (, ) v v u u f u v du dv F.

7 Eample.6(b) ad (c) 7 Solutio Fom.9 the coditioal p.d.f of give is deoted b f ( ) ad defied as f ( ) f ( ) ( ) f ad ad f ( ) is the magial p.d.f. of. We kow f ( ) fid f ( ). Thee ae two was ou ca fid f ( ) f whee f (, ) is the joit p.d.f. of, 8 7 so we eed to. The fist wa ivolves itegatio ad the secod wa ivolves diffeetiatio. I will do both was to show ou how to use the diffeet esults we have hee but ou should alwas choose the wa ou fid easiest i.e ou would ot be epected to fid f ( ) both was i a assessed wok. Method Fom.8 f ( ) f (, ) d so f ( ) 8 7 d 8 7 d Method Fom. f ( ) df ( ) d Fom.0 F ( ) F( ) F F whee F ( ) MA, whee MA ( ) F ad fom pat (a), F( ) ( ) Hece f ( ) 4 4, ( ) d 4 d 4 is the magial c.d.f of. is the lagest value of, so 4, ( ) 8 hece F ( ) so 4. as with method. Now theefoe the coditioal desit fuctio of give, f ( ) is give b So ( ) f f ( ) (, ) f ( ) f 7 ad 0 othewise (c) Now f ( ) is a fuctio of ol, so usig esult.(c), ad ae idepedet. Notice also that f ( ) f ( ) which ou would epect if ad ae idepedet.

8 . Epectatios ad vaiaces 8 Discete adom vaiables ( ) p(, ) p ( ) E p ( ) p(, ) p ( ) E p Eamples,...,... Hece Va() E( ) ( E( )), Va() E( ) ( E( ) Cotiuous adom vaiables ( ) ( ) ( ) etc. E f, d d f d,... ( ) ( ) ( ) E f, d d f d,... Eamples.4 Epectatio of a fuctio of the.v.'s ad Cotiuous ad Discete ad E[ g] g f dd e. g. E f dd E[] f (, ) dd..5 Covaiace ad coelatio Covaiace of ad is defied as follows : Cov (,) σ E() - E()E(). Notes (a) If the adom vaiables icease togethe o decease togethe, the the covaiace will be positive, wheeas if oe adom vaiable iceases ad the othe vaiable deceases ad vice-vesa, the the covaiace will be egative. (b) If ad ae idepedet.v's, the E() E()E() so cov(, ) 0. Howeve if cov(,) 0, it does ot follow that ad ae idepedet uless ad ae

9 9 Nomal.v's. Coelatio coefficiet ρ co(,) Cov (, ). σ σ Note (a) The coelatio coefficiet is a umbe betwee - ad i.e. - ρ (b) If the adom vaiables icease togethe o decease togethe, the ρ will be positive, wheeas if oe adom vaiable iceases ad the othe vaiable deceases ad vice-vesa, the ρ will be egative. (c) It measues the degee of liea elatioship betwee the two adom vaiables ad, so if thee is a o-liea elatioship betwee ad o ad ae idepedet adom vaiables, the ρ will be 0. ou will stud coelatio i moe detail i the Ecoometic pat of the couse with David Wite. Eample.7 I Eample. ae ad coelated? Solutio Below is the joit o bivaiate pobabilit distibutio of ad : The magial distibutios of ad ae Total p P( ) o ( ) ad P( ) o ( ) 0 0 Total p Eample. 8 I Eample.6 (i) Calculate E(), Va(), E() ad cov(,). (ii) Ae ad idepedet?.4 Useful esults o epectatios ad vaiaces (i) E( a + b) ae( ) + be( ) whee a ad b ae costats.

10 0 (ii) Va( a + b) a Va( ) + b Va( ) + ab cov. Result (i) ca be eteded to a adom vaiables,,..., E a + a a a E + a E a E ( ) ( ) ( ) ( ) Whe ad ae idepedet, the (iii) Va( a + b) a Va( ) + b Va( ) so cov(, ) 0 (iv) E( ) E( ) E( ) Results (iii) ad (iv) ca be eteded to a idepedet adom vaiables,,..., (iii)* Va( a a... a ) ( ) + ( ) ( ) a Va a Va a Va (iv)* E(... ) E( ). E( )... E( )

11 .5 Combiatios of idepedet Nomal adom vaiables Suppose ~ N ( µ, σ ), ~ N( µ, σ ), ~ N ( µ, σ ),...ad ~ N ( µ, σ ),,..., ae idepedet adom vaiables, the if a + a + a a whee a, a... a ae costats, ~ N( a µ + a µ + a µ a µ, a σ + a σ a σ ) i.e. ~ N( a µ, a σ ). i i i i

12 I paticula, suppose,... fom a adom sample fom a Nomal populatio with mea µ ad vaiace σ, µ µ µ... µ µ ad σ σ... σ σ. ~ N( a µ, a σ ). i i Futhe, suppose that a a a... a the ad ~ N( µ, σ ).

Chi-squared goodness-of-fit test.

Chi-squared goodness-of-fit test. Sectio 1 Chi-squaed goodess-of-fit test. Example. Let us stat with a Matlab example. Let us geeate a vecto X of 1 i.i.d. uifom adom vaiables o [, 1] : X=ad(1,1). Paametes (1, 1) hee mea that we geeate

More information

BINOMIAL THEOREM. 1. Introduction. 2. The Binomial Coefficients. ( x + 1), we get. and. When we expand

BINOMIAL THEOREM. 1. Introduction. 2. The Binomial Coefficients. ( x + 1), we get. and. When we expand BINOMIAL THEOREM Itoductio Whe we epad ( + ) ad ( + ), we get ad ( + ) = ( + )( + ) = + + + = + + ( + ) = ( + )( + ) = ( + )( + + ) = + + + + + = + + + 4 5 espectively Howeve, whe we ty to epad ( + ) ad

More information

Periodic Review Probabilistic Multi-Item Inventory System with Zero Lead Time under Constraints and Varying Order Cost

Periodic Review Probabilistic Multi-Item Inventory System with Zero Lead Time under Constraints and Varying Order Cost Ameica Joual of Applied Scieces (8: 3-7, 005 ISS 546-939 005 Sciece Publicatios Peiodic Review Pobabilistic Multi-Item Ivetoy System with Zeo Lead Time ude Costaits ad Vayig Ode Cost Hala A. Fegay Lectue

More information

Chapter 2 Sequences and Series

Chapter 2 Sequences and Series Chapte 7 Sequece ad seies Chapte Sequeces ad Seies. Itoductio: The INVENTOR of chess asked the Kig of the Kigdom that he may be ewaded i lieu of his INVENTION with oe gai of wheat fo the fist squae of

More information

Two degree of freedom systems. Equations of motion for forced vibration Free vibration analysis of an undamped system

Two degree of freedom systems. Equations of motion for forced vibration Free vibration analysis of an undamped system wo degee of feedom systems Equatios of motio fo foced vibatio Fee vibatio aalysis of a udamped system Itoductio Systems that equie two idepedet d coodiates to descibe thei motio ae called two degee of

More information

Equation of a line. Line in coordinate geometry. Slope-intercept form ( 斜 截 式 ) Intercept form ( 截 距 式 ) Point-slope form ( 點 斜 式 )

Equation of a line. Line in coordinate geometry. Slope-intercept form ( 斜 截 式 ) Intercept form ( 截 距 式 ) Point-slope form ( 點 斜 式 ) Chapter : Liear Equatios Chapter Liear Equatios Lie i coordiate geometr I Cartesia coordiate sstems ( 卡 笛 兒 坐 標 系 統 ), a lie ca be represeted b a liear equatio, i.e., a polomial with degree. But before

More information

Understanding Financial Management: A Practical Guide Guideline Answers to the Concept Check Questions

Understanding Financial Management: A Practical Guide Guideline Answers to the Concept Check Questions Udestadig Fiacial Maagemet: A Pactical Guide Guidelie Aswes to the Cocept Check Questios Chapte 4 The Time Value of Moey Cocept Check 4.. What is the meaig of the tems isk-etu tadeoff ad time value of

More information

Chapter 9: Correlation and Regression: Solutions

Chapter 9: Correlation and Regression: Solutions Chapter 9: Correlatio ad Regressio: Solutios 9.1 Correlatio I this sectio, we aim to aswer the questio: Is there a relatioship betwee A ad B? Is there a relatioship betwee the umber of emploee traiig hours

More information

THE PRINCIPLE OF THE ACTIVE JMC SCATTERER. Seppo Uosukainen

THE PRINCIPLE OF THE ACTIVE JMC SCATTERER. Seppo Uosukainen THE PRINCIPLE OF THE ACTIVE JC SCATTERER Seppo Uoukaie VTT Buildig ad Tapot Ai Hadlig Techology ad Acoutic P. O. Bo 1803, FIN 02044 VTT, Filad Seppo.Uoukaie@vtt.fi ABSTRACT The piciple of fomulatig the

More information

Section 11.3: The Integral Test

Section 11.3: The Integral Test Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult

More information

Infinite Sequences and Series

Infinite Sequences and Series CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...

More information

I. Chi-squared Distributions

I. Chi-squared Distributions 1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

More information

Soving Recurrence Relations

Soving Recurrence Relations Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree

More information

The second difference is the sequence of differences of the first difference sequence, 2

The second difference is the sequence of differences of the first difference sequence, 2 Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for

More information

Paper SD-07. Key words: upper tolerance limit, macros, order statistics, sample size, confidence, coverage, binomial

Paper SD-07. Key words: upper tolerance limit, macros, order statistics, sample size, confidence, coverage, binomial SESUG 212 Pae SD-7 Samle Size Detemiatio fo a Noaametic Ue Toleace Limit fo ay Ode Statistic D. Deis Beal, Sciece Alicatios Iteatioal Cooatio, Oak Ridge, Teessee ABSTRACT A oaametic ue toleace limit (UTL)

More information

Joint Probability Distributions and Random Samples

Joint Probability Distributions and Random Samples STAT5 Sprig 204 Lecture Notes Chapter 5 February, 204 Joit Probability Distributios ad Radom Samples 5. Joitly Distributed Radom Variables Chapter Overview Joitly distributed rv Joit mass fuctio, margial

More information

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5 Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.

More information

Hypothesis testing. Null and alternative hypotheses

Hypothesis testing. Null and alternative hypotheses Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate

More information

Normal Distribution.

Normal Distribution. Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued

More information

Derivation of Annuity and Perpetuity Formulae. A. Present Value of an Annuity (Deferred Payment or Ordinary Annuity)

Derivation of Annuity and Perpetuity Formulae. A. Present Value of an Annuity (Deferred Payment or Ordinary Annuity) Aity Deivatios 4/4/ Deivatio of Aity ad Pepetity Fomlae A. Peset Vale of a Aity (Defeed Paymet o Odiay Aity 3 4 We have i the show i the lecte otes ad i ompodi ad Discoti that the peset vale of a set of

More information

Mechanics 1: Motion in a Central Force Field

Mechanics 1: Motion in a Central Force Field Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.

More information

On Correlation Coefficient. The correlation coefficient indicates the degree of linear dependence of two random variables.

On Correlation Coefficient. The correlation coefficient indicates the degree of linear dependence of two random variables. C.Candan EE3/53-METU On Coelation Coefficient The coelation coefficient indicates the degee of linea dependence of two andom vaiables. It is defined as ( )( )} σ σ Popeties: 1. 1. (See appendi fo the poof

More information

Learning Objectives. Chapter 2 Pricing of Bonds. Future Value (FV)

Learning Objectives. Chapter 2 Pricing of Bonds. Future Value (FV) Leaig Objectives Chapte 2 Picig of Bods time value of moey Calculate the pice of a bod estimate the expected cash flows detemie the yield to discout Bod pice chages evesely with the yield 2-1 2-2 Leaig

More information

Properties of MLE: consistency, asymptotic normality. Fisher information.

Properties of MLE: consistency, asymptotic normality. Fisher information. Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout

More information

Notes on Hypothesis Testing

Notes on Hypothesis Testing Probability & Statistics Grishpa Notes o Hypothesis Testig A radom sample X = X 1,..., X is observed, with joit pmf/pdf f θ x 1,..., x. The values x = x 1,..., x of X lie i some sample space X. The parameter

More information

1 Computing the Standard Deviation of Sample Means

1 Computing the Standard Deviation of Sample Means Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.

More information

4.1 Sigma Notation and Riemann Sums

4.1 Sigma Notation and Riemann Sums 0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

More information

The dinner table problem: the rectangular case

The dinner table problem: the rectangular case The ie table poblem: the ectagula case axiv:math/009v [mathco] Jul 00 Itouctio Robeto Tauaso Dipatimeto i Matematica Uivesità i Roma To Vegata 00 Roma, Italy tauaso@matuiomait Decembe, 0 Assume that people

More information

2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES

2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES . TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an

More information

Section IV.5: Recurrence Relations from Algorithms

Section IV.5: Recurrence Relations from Algorithms Sectio IV.5: Recurrece Relatios from Algorithms Give a recursive algorithm with iput size, we wish to fid a Θ (best big O) estimate for its ru time T() either by obtaiig a explicit formula for T() or by

More information

1 The Binomial Theorem: Another Approach

1 The Binomial Theorem: Another Approach The Biomial Theorem: Aother Approach Pascal s Triagle I class (ad i our text we saw that, for iteger, the biomial theorem ca be stated (a + b = c a + c a b + c a b + + c ab + c b, where the coefficiets

More information

Chapter 5 Discrete Probability Distributions

Chapter 5 Discrete Probability Distributions Slides Prepared by JOHN S. LOUCKS St. Edward s Uiversity Slide Chapter 5 Discrete Probability Distributios Radom Variables Discrete Probability Distributios Epected Value ad Variace Poisso Distributio

More information

Semipartial (Part) and Partial Correlation

Semipartial (Part) and Partial Correlation Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated

More information

Skills Needed for Success in Calculus 1

Skills Needed for Success in Calculus 1 Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell

More information

ECE 340 Lecture 13 : Optical Absorption and Luminescence 2/19/14 ( ) Class Outline: Band Bending Optical Absorption

ECE 340 Lecture 13 : Optical Absorption and Luminescence 2/19/14 ( ) Class Outline: Band Bending Optical Absorption /9/4 ECE 34 Lectue 3 : Optical Absoptio ad Lumiescece Class Outlie: Thigs you should kow whe you leave Key Questios How do I calculate kietic ad potetial eegy fom the bads? What is diect ecombiatio? How

More information

Coordinate Systems L. M. Kalnins, March 2009

Coordinate Systems L. M. Kalnins, March 2009 Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean

More information

Annuities and loan. repayments. Syllabus reference Financial mathematics 5 Annuities and loan. repayments

Annuities and loan. repayments. Syllabus reference Financial mathematics 5 Annuities and loan. repayments 8 8A Futue value of a auity 8B Peset value of a auity 8C Futue ad peset value tables 8D Loa epaymets Auities ad loa epaymets Syllabus efeece Fiacial mathematics 5 Auities ad loa epaymets Supeauatio (othewise

More information

Sampling Distribution And Central Limit Theorem

Sampling Distribution And Central Limit Theorem () Samplig Distributio & Cetral Limit Samplig Distributio Ad Cetral Limit Samplig distributio of the sample mea If we sample a umber of samples (say k samples where k is very large umber) each of size,

More information

THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n

THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample

More information

Heat (or Diffusion) equation in 1D*

Heat (or Diffusion) equation in 1D* Heat (or Diffusio) equatio i D* Derivatio of the D heat equatio Separatio of variables (refresher) Worked eamples *Kreysig, 8 th Ed, Sectios.4b Physical assumptios We cosider temperature i a log thi wire

More information

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008 I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

More information

Functions of a Random Variable: Density. Math 425 Intro to Probability Lecture 30. Definition Nice Transformations. Problem

Functions of a Random Variable: Density. Math 425 Intro to Probability Lecture 30. Definition Nice Transformations. Problem Intoduction One Function of Random Vaiables Functions of a Random Vaiable: Density Math 45 Into to Pobability Lectue 30 Let gx) = y be a one-to-one function whose deiatie is nonzeo on some egion A of the

More information

Mechanics 1: Work, Power and Kinetic Energy

Mechanics 1: Work, Power and Kinetic Energy Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).

More information

1. MATHEMATICAL INDUCTION

1. MATHEMATICAL INDUCTION 1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1

More information

Asymptotic Growth of Functions

Asymptotic Growth of Functions CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll

More information

Gauss Law. Physics 231 Lecture 2-1

Gauss Law. Physics 231 Lecture 2-1 Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing

More information

Chapter 14 Nonparametric Statistics

Chapter 14 Nonparametric Statistics Chapter 14 Noparametric Statistics A.K.A. distributio-free statistics! Does ot deped o the populatio fittig ay particular type of distributio (e.g, ormal). Sice these methods make fewer assumptios, they

More information

Module 4: Mathematical Induction

Module 4: Mathematical Induction Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate

More information

Geometric Sequences. Definition: A geometric sequence is a sequence of the form

Geometric Sequences. Definition: A geometric sequence is a sequence of the form Geometic equeces Aothe simple wy of geetig sequece is to stt with umbe d epetedly multiply it by fixed ozeo costt. This type of sequece is clled geometic sequece. Defiitio: A geometic sequece is sequece

More information

Overview of some probability distributions.

Overview of some probability distributions. Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability

More information

7. Sample Covariance and Correlation

7. Sample Covariance and Correlation 1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y

More information

S. Tanny MAT 344 Spring 1999. be the minimum number of moves required.

S. Tanny MAT 344 Spring 1999. be the minimum number of moves required. S. Tay MAT 344 Sprig 999 Recurrece Relatios Tower of Haoi Let T be the miimum umber of moves required. T 0 = 0, T = 7 Iitial Coditios * T = T + $ T is a sequece (f. o itegers). Solve for T? * is a recurrece,

More information

University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution

University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chi-square (χ ) distributio.

More information

Solving Divide-and-Conquer Recurrences

Solving Divide-and-Conquer Recurrences Solvig Divide-ad-Coquer Recurreces Victor Adamchik A divide-ad-coquer algorithm cosists of three steps: dividig a problem ito smaller subproblems solvig (recursively) each subproblem the combiig solutios

More information

Chapter 4 Multivariate distributions

Chapter 4 Multivariate distributions Chapter 4 Multivariate distributios k Multivariate Distributios All the results derived for the bivariate case ca be geeralized to RV. The joit CDF of,,, k will have the form: P(x, x,, x k ) whe the RVs

More information

Fourier Series and the Wave Equation Part 2

Fourier Series and the Wave Equation Part 2 Fourier Series ad the Wave Equatio Part There are two big ideas i our work this week. The first is the use of liearity to break complicated problems ito simple pieces. The secod is the use of the symmetries

More information

Trigonometric Functions of Any Angle

Trigonometric Functions of Any Angle Tigonomet Module T2 Tigonometic Functions of An Angle Copight This publication The Nothen Albeta Institute of Technolog 2002. All Rights Reseved. LAST REVISED Decembe, 2008 Tigonometic Functions of An

More information

CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations

CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad

More information

Partial Di erential Equations

Partial Di erential Equations Partial Di eretial Equatios Partial Di eretial Equatios Much of moder sciece, egieerig, ad mathematics is based o the study of partial di eretial equatios, where a partial di eretial equatio is a equatio

More information

UC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006

UC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006 Exam format UC Bereley Departmet of Electrical Egieerig ad Computer Sciece EE 6: Probablity ad Radom Processes Solutios 9 Sprig 006 The secod midterm will be held o Wedesday May 7; CHECK the fial exam

More information

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value

More information

Maximum Likelihood Estimators.

Maximum Likelihood Estimators. Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio

More information

, a Wishart distribution with n -1 degrees of freedom and scale matrix.

, a Wishart distribution with n -1 degrees of freedom and scale matrix. UMEÅ UNIVERSITET Matematisk-statistiska istitutioe Multivariat dataaalys D MSTD79 PA TENTAMEN 004-0-9 LÖSNINGSFÖRSLAG TILL TENTAMEN I MATEMATISK STATISTIK Multivariat dataaalys D, 5 poäg.. Assume that

More information

92.131 Calculus 1 Optimization Problems

92.131 Calculus 1 Optimization Problems 9 Calculus Optimization Poblems ) A Noman window has the outline of a semicicle on top of a ectangle as shown in the figue Suppose thee is 8 + π feet of wood tim available fo all 4 sides of the ectangle

More information

Counting Principles and Generating Functions

Counting Principles and Generating Functions Uit- Coutig Piciples ad Geeatig Fuctios. THE RULES OF SUM AND PRODUCT.. Sum Rule. (The Piciple of disjuctive Coutig) If a fist task ca be doe i ways ad a secod task i ways, ad if these tasks caot be doe

More information

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval

More information

Psychology 282 Lecture #2 Outline. Review of Pearson correlation coefficient:

Psychology 282 Lecture #2 Outline. Review of Pearson correlation coefficient: Psychology 282 Lectue #2 Outline Review of Peason coelation coefficient: z z ( n 1) Measue of linea elationship. Magnitude Stength Sign Diection Bounded by +1.0 and -1.0. Independent of scales of measuement.

More information

Maximum Entropy, Parallel Computation and Lotteries

Maximum Entropy, Parallel Computation and Lotteries Maximum Etopy, Paallel Computatio ad Lotteies S.J. Cox Depatmet of Electoics ad Compute Sciece, Uivesity of Southampto, UK. G.J. Daiell Depatmet of Physics ad Astoomy, Uivesity of Southampto, UK. D.A.

More information

Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series

Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series 8 Fourier Series Our aim is to show that uder reasoable assumptios a give -periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series

More information

Trigonometry in the Cartesian Plane

Trigonometry in the Cartesian Plane Tigonomet in the Catesian Plane CHAT Algeba sec. 0. to 0.5 *Tigonomet comes fom the Geek wod meaning measuement of tiangles. It pimail dealt with angles and tiangles as it petained to navigation astonom

More information

2.2. Trigonometric Ratios of Any Angle. Investigate Trigonometric Ratios for Angles Greater Than 90

2.2. Trigonometric Ratios of Any Angle. Investigate Trigonometric Ratios for Angles Greater Than 90 . Tigonometic Ratios of An Angle Focus on... detemining the distance fom the oigin to a point (, ) on the teminal am of an angle detemining the value of sin, cos, o tan given an point (, ) on the teminal

More information

Laminar and non-laminar flow in geosynthetic and granular drains

Laminar and non-laminar flow in geosynthetic and granular drains Geosythetics Iteatioal, 2012, 19, No. 2 Lamia ad o-lamia flow i geosythetic ad gaula dais J. P. Gioud 1, J. P. Gouc 2 ad E. Kavazajia, J. 1 Cosultig Egiee, JP Gioud, Ic., 587 Noth Ocea Blvd, Ocea Ridge,

More information

Your organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:

Your organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows: Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network

More information

Finance Practice Problems

Finance Practice Problems Iteest Fiace Pactice Poblems Iteest is the cost of boowig moey. A iteest ate is the cost stated as a pecet of the amout boowed pe peiod of time, usually oe yea. The pevailig maket ate is composed of: 1.

More information

Economics 326: Input Demands. Ethan Kaplan

Economics 326: Input Demands. Ethan Kaplan Economics 326: Input Demands Ethan Kaplan Octobe 24, 202 Outline. Tems 2. Input Demands Tems Labo Poductivity: Output pe unit of labo. Y (K; L) L What is the labo poductivity of the US? Output is ouhgly

More information

THE GEOMETRIC SERIES

THE GEOMETRIC SERIES Mthemtics Revisio Guides The Geometic eies Pge of M.K. HOME TUITION Mthemtics Revisio Guides Level: A / A Level AQA : C Edexcel: C OCR: C OCR MEI: C THE GEOMETRIC ERIE Vesio :. Dte: 8-06-0 Exmples 7 d

More information

An example of non-quenched convergence in the conditional central limit theorem for partial sums of a linear process

An example of non-quenched convergence in the conditional central limit theorem for partial sums of a linear process A example of o-queched covergece i the coditioal cetral limit theorem for partial sums of a liear process Dalibor Volý ad Michael Woodroofe Abstract A causal liear processes X,X 0,X is costructed for which

More information

Key Ideas Section 8-1: Overview hypothesis testing Hypothesis Hypothesis Test Section 8-2: Basics of Hypothesis Testing Null Hypothesis

Key Ideas Section 8-1: Overview hypothesis testing Hypothesis Hypothesis Test Section 8-2: Basics of Hypothesis Testing Null Hypothesis Chapter 8 Key Ideas Hypothesis (Null ad Alterative), Hypothesis Test, Test Statistic, P-value Type I Error, Type II Error, Sigificace Level, Power Sectio 8-1: Overview Cofidece Itervals (Chapter 7) are

More information

Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:

Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas: Chapter 7 - Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries

More information

Chapter 3 Savings, Present Value and Ricardian Equivalence

Chapter 3 Savings, Present Value and Ricardian Equivalence Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,

More information

Breakeven Holding Periods for Tax Advantaged Savings Accounts with Early Withdrawal Penalties

Breakeven Holding Periods for Tax Advantaged Savings Accounts with Early Withdrawal Penalties Beakeve Holdig Peiods fo Tax Advataged Savigs Accouts with Ealy Withdawal Pealties Stephe M. Hoa Depatmet of Fiace St. Boavetue Uivesity St. Boavetue, New Yok 4778 Phoe: 76-375-209 Fax: 76-375-29 e-mail:

More information

Case Study. Normal and t Distributions. Density Plot. Normal Distributions

Case Study. Normal and t Distributions. Density Plot. Normal Distributions Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca

More information

The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles

The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio

More information

Chapter 7 Methods of Finding Estimators

Chapter 7 Methods of Finding Estimators Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of

More information

Chapter 5: Inner Product Spaces

Chapter 5: Inner Product Spaces Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples

More information

USING STATISTICAL FUNCTIONS ON A SCIENTIFIC CALCULATOR

USING STATISTICAL FUNCTIONS ON A SCIENTIFIC CALCULATOR USING STATISTICAL FUNCTIONS ON A SCIENTIFIC CALCULATOR Objective:. Improve calculator skills eeded i a multiple choice statistical eamiatio where the eam allows the studet to use a scietific calculator..

More information

Simulation and Monte Carlo integration

Simulation and Monte Carlo integration Chapter 3 Simulatio ad Mote Carlo itegratio I this chapter we itroduce the cocept of geeratig observatios from a specified distributio or sample, which is ofte called Mote Carlo geeratio. The ame of Mote

More information

ARITHMETIC AND GEOMETRIC PROGRESSIONS

ARITHMETIC AND GEOMETRIC PROGRESSIONS Arithmetic Ad Geometric Progressios Sequeces Ad ARITHMETIC AND GEOMETRIC PROGRESSIONS Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives

More information

3. Covariance and Correlation

3. Covariance and Correlation Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics

More information

Power and Sample Size Calculations for the 2-Sample Z-Statistic

Power and Sample Size Calculations for the 2-Sample Z-Statistic Powe and Sample Size Calculations fo the -Sample Z-Statistic James H. Steige ovembe 4, 004 Topics fo this Module. Reviewing Results fo the -Sample Z (a) Powe and Sample Size in Tems of a oncentality Paamete.

More information

Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.

Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL. Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 3000, Haifa, Israel I memory

More information

Section 3.3: Geometric Sequences and Series

Section 3.3: Geometric Sequences and Series ectio 3.3: Geometic equeces d eies Geometic equeces Let s stt out with defiitio: geometic sequece: sequece i which the ext tem is foud by multiplyig the pevious tem by costt (the commo tio ) Hee e some

More information

Questions for Review. By buying bonds This period you save s, next period you get s(1+r)

Questions for Review. By buying bonds This period you save s, next period you get s(1+r) MACROECONOMICS 2006 Week 5 Semina Questions Questions fo Review 1. How do consumes save in the two-peiod model? By buying bonds This peiod you save s, next peiod you get s() 2. What is the slope of a consume

More information

A probabilistic proof of a binomial identity

A probabilistic proof of a binomial identity A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two

More information

AN APPROXIMATE ISOPERIMETRIC INEQUALITY FOR r-sets

AN APPROXIMATE ISOPERIMETRIC INEQUALITY FOR r-sets AN APPROXIMATE ISOPERIMETRIC INEQUALITY FOR -SETS DEMETRES CHRISTOFIDES, DAVID ELLIS AND PETER KEEVASH Abstact. We pove a vetex-isopeimetic iequality fo [], the set of all -elemet subsets of {1, 2,...,

More information

Graphs of Equations. A coordinate system is a way to graphically show the relationship between 2 quantities.

Graphs of Equations. A coordinate system is a way to graphically show the relationship between 2 quantities. Gaphs of Equations CHAT Pe-Calculus A coodinate sstem is a wa to gaphicall show the elationship between quantities. Definition: A solution of an equation in two vaiables and is an odeed pai (a, b) such

More information

Distributions of Order Statistics

Distributions of Order Statistics Chapter 2 Distributios of Order Statistics We give some importat formulae for distributios of order statistics. For example, where F k: (x)=p{x k, x} = I F(x) (k, k + 1), I x (a,b)= 1 x t a 1 (1 t) b 1

More information

f(x + T ) = f(x), for all x. The period of the function f(t) is the interval between two successive repetitions.

f(x + T ) = f(x), for all x. The period of the function f(t) is the interval between two successive repetitions. Fourier Series. Itroductio Whe the Frech mathematicia Joseph Fourier (768-83) was tryig to study the flow of heat i a metal plate, he had the idea of expressig the heat source as a ifiite series of sie

More information

Section 9.2 Series and Convergence

Section 9.2 Series and Convergence Sectio 9. Series ad Covergece Goals of Chapter 9 Approximate Pi Prove ifiite series are aother importat applicatio of limits, derivatives, approximatio, slope, ad cocavity of fuctios. Fid challegig atiderivatives

More information

Page 2 of 14 = T(-2) + 2 = [ T(-3)+1 ] + 2 Substitute T(-3)+1 for T(-2) = T(-3) + 3 = [ T(-4)+1 ] + 3 Substitute T(-4)+1 for T(-3) = T(-4) + 4 After i

Page 2 of 14 = T(-2) + 2 = [ T(-3)+1 ] + 2 Substitute T(-3)+1 for T(-2) = T(-3) + 3 = [ T(-4)+1 ] + 3 Substitute T(-4)+1 for T(-3) = T(-4) + 4 After i Page 1 of 14 Search C455 Chapter 4 - Recursio Tree Documet last modified: 02/09/2012 18:42:34 Uses: Use recursio tree to determie a good asymptotic boud o the recurrece T() = Sum the costs withi each level

More information